How to multiply variables - ISEE Upper Level Mathematics Achievement

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Question

Machine A can produce 4 buttons in 2 days. Machine B can produce 20 buttons in 4 days. How many buttons can Machine A and B, working together, produce in 40 days?

Answer

First determine how many buttons each machine can produce in a day.

Machine A: $\dpi{100} \frac{4}{2}=2\ buttons\ per\ day$

Machine B: $\dpi{100} \frac{20}{4}=5\ buttons\ per\ day$

Machine A and B can produce 7 buttons per day when they are working together.

$\dpi{100} 7\times 40=280\ buttons$

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Question

Brian has 3 siblings. When his family orders pizza, each of the 4 children is given $\dpi{100} \frac{1}{4}$ of the pizza. Brian does not feel well so he only finishes $\dpi{100} \frac{1}{3}$ of his pizza. If the original pizza consisted of 12 slices of pizza, how many slices did Brian eat?

Answer

Brian eats $\dpi{100} \frac{1}{3}\times \frac{1}{4}\times 12\ slices\ of\ pizza$ .

So Brian eats

$\dpi{100} \frac{1}{12}\cdot 12=1\ slice\ of\ pizza.$

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Question

Solve for $\dpi{100} x$ :

$\dpi{100} \frac{1}{3}x-14=7$

Answer

$\dpi{100} \frac{1}{3}x-14=7$

$\dpi{100} \frac{1}{3}x-14+14=7+14$

$\dpi{100} \frac{1}{3}x=21$

$\dpi{100} 3\cdot \frac{1}{3}x=21\cdot 3$

$\dpi{100} x=63$

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Question

Solve for $\dpi{100} x$ :

$\dpi{100} 4x^{2}=256$

Answer

$\dpi{100} 4x^{2}=256$

$\dpi{100} \frac{4x^{2}}{4}=\frac{256}{4}$

$\dpi{100} x^{2}=64$

$\dpi{100} \sqrt{x^{2}}=\sqrt{64}$

$\dpi{100} x=\pm 8$

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Question

Simplify:

$(0.6x-0.4y) ^{2}$

Answer

This can be solved using the pattern for the square of a difference:

$(0.6x-0.4y) ^{2}$

$=\left ( 0.6x \right ) ^{2} - 2\cdot 0.6x \cdot 0.4y + \left ( 0.4y \right ) ^{2}$

$= 0.6^{2}x^{2} - 2\cdot 0.6 \cdot 0.4 \cdot xy + 0.4^{2} y ^{2}$

$= 0.36x^{2} - 0.48 xy + 0.16 y ^{2}$

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Question

Simplify:

$(0.6x+0.4y) ^{2}$

Answer

This can be solved using the pattern for the square of a sum:

$(0.6x+0.4y) ^{2}$

$=\left ( 0.6x \right ) ^{2} +2\cdot 0.6x \cdot 0.4y + \left ( 0.4y \right ) ^{2}$

$= 0.6^{2}x^{2} +2\cdot 0.6 \cdot 0.4 \cdot xy + 0.4^{2} y ^{2}$

$= 0.36x^{2} + 0.48 xy + 0.16 y ^{2}$

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Question

Simplify:

$\left (\frac{4}{3}x + \frac{5}{3} \right )^{2}$

Answer

This can be solved using the pattern for the square of a sum:

$\left (\frac{4}{3}x + \frac{5}{3} \right )^{2}$

$=\left (\frac{4}{3}x \right ) ^{2} +2 \cdot \frac{4}{3}x \cdot \frac{5}{3} + \left ( \frac{5}{3} \right ) ^{2}$

$=\frac{4^{2}}{3^{2}}\cdot x^{2} +2 \cdot \frac{4}{3}\cdot \frac{5}{3} \cdot x +\frac{5^{2}}{3^{2}}$

$=\frac{16}{9}x^{2} + \frac{40}{9}x +\frac{25}{9}$

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Question

Multiply:

Answer

Use the FOIL method:

$= 5x \cdot 3x-5x \cdot 8y - 3y \cdot 3x + 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x-5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x + 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}-40xy - 9xy +24 y^{2}$

$= 15 x ^{2}-49xy +24 y^{2}$

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Question

Multiply:

Answer

$= 5x \cdot 3x+5x \cdot 8y - 3y \cdot 3x - 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x+5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x - 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}+40xy - 9xy -24 y^{2}$

$= 15 x ^{2}+31xy -24 y^{2}$

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Question

Define

What is ?

Answer

Substitute for :

$f(3-2x) = 5 - 2 \left ( 3-2x \right )$

$f(3-2x) = 5 - 2 \cdot 3- \left ( - 2 \right )\cdot 2x \right )$

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Question

Simplify:

$(\frac{4x^2}{81x^6})^{-\frac{1}{2}}$

Answer

First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

$(\frac{4x^2}{81x^6})^{-\frac{1}{2}}=(\frac{81x^6}{4x^2})^{\frac{1}{2}}$

Apply the exponent within the parentheses and simplify. Remember that fractional exponents can be written as roots.

$\frac{(81x^6)^{\frac{1}{2}}}{(4x^2)^{\frac{1}{2}}}=\frac{\sqrt{81x^6}}{\sqrt{4x^2}}$

Simplify by taking the roots and canceling common factors.

$\frac{\sqrt{81x^6}}{\sqrt{4x^2}}=\frac{9x^3}{2x}=\frac{9x^2}{2}$

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Question

Simplify:

$(-\frac{5}{3})^{-3}$

Answer

First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

$(-\frac{5}{3})^{-3}=(-\frac{3}{5})^3$

Apply the exponent within the parentheses and simplify. The negative in the fraction can be applied to either the numerator or the denominator, but not both; we will apply it to the numerator.

$\frac{(-3)^3}{(5)^3}$

$\frac{-27}{125}=-\frac{27}{125}$

The fraction cannot be simplified further.

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Question

Simplify:

$4xy^2-2xy+y^{2}x+y^3-8xy+4yx$

Answer

Reorder the expression to group like-terms together.

$4xy^2-2xy+y^{2}x+y^3-8xy+4yx$

$(4xy^2+y^{2}x)+(-2xy+-8xy+4yx)+y^3$

Simplify by combining like-terms.

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Question

If the product of three consecutive numbers is 990, what is the smallest number?

Answer

If the product of three consecutive numbers is 990, then the smallest number can be found by plugging in each answer choice into the scenario to see whether it is correct.

If we plug in 9 as the smallest number, then the two consecutive numbers would be 10 and 11.

Given that 9 times 10 times 11 equals 990, that is the correct answer.

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Question

Megan teaches a cooking class. Every time a student takes a class, the student receives a 10% discount off of the price he paid for the previous class. The first class that Jose took cost $50. What will be the first class in which Jose pays less than $38?

Answer

In order to solve this problem, the price of $50 must be discounted by 10% until we get to a price of $38 or less.

The first class is 50.

The second class, being 10 percent less than the previous one, is $50-.1\cdot 50=50-5=45$

The third class, being 10 percent less than the previous one, is $45-.1\cdot45=45-4.5 = 40.5$

The fourth class, being 10 percent less than the previous one, is $40.5-.1\cdot40.5=40.5-4.05=36.45$

Therefore, the answer is 4.

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Question

If $x^{2}-12x$ is a positive number, what is a possible value of ?

Answer

If x is equal to , then the equation could be written as follows:

$x^{2}-12x$

$-4^{2}-(12*-4)$

Given that is a positive number, is a possible value of .

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Question

What is the simplified version of the expression below?

$(\frac{5}{4}\cdot8+6\div3)^{2}-12$

Answer

The first step is to simplify the values in the parentheses:

$(\frac{5}{4}\cdot8+6\div3)^{2}-12$

$(10+2)^{2}-12$

$12^{2}-12$

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Question

$x^{2} = 18$

$y^{2} = 2$

and are both positive.

Evaluate .

Answer

Multiply the binomials using the FOIL method - first, outer, inner, last - then combine like terms:

$= 2x \cdot 3x - 2x \cdot 2y+ 3y \cdot 3x-3y \cdot 2y$

$=6 x ^{2}-4 x y+ 9xy-6y^{2}$

$=6 x ^{2}+ 5 xy-6y^{2}$

$x^{2} = 18$ and $y^{2} = 2$ ; also, by the Power of a Product Principle:

$(xy) ^{2} = x^{2}y^{2}$ .

and are both positive, so, substituting:

$xy = \sqrt{x^{2}y^{2}} = \sqrt{18 \cdot 2} = \sqrt{36} = 6$ .

Again, using substitution:

$=6 x ^{2}+ 5 xy-6y^{2}$

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Question

Simplify the following expression

Answer

Simplify the following expression

Let's begin by multiplying our coefficients:

Next, we need to realize that we can combine our x's by adding the exponents.

$x^2*x^3=x^{3+2}=x^5$

Put it all together to get:

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Question

Simplify the following:

Answer

Simplify the following:

To begin, our coefficient will not change. We have just one integer (the 9) and nothing to multiply it by.

To combine our exponents, we will add them. This is because we are multiplying them

$q^3q^5=q^{3+5}=q^8$

$n^3n^6=n^{3+6}=n^9$

Put it together to get:

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